So what is the difference between harmonic relative pitch (that Interval Loader teaches) and regular relative pitch? If I learned harmonic relative pitch from Interval Loader, would I need another program to learn regular relative pitch as well?

Essentially, the point can be stated that "distance" between notes is just a metaphor, and not actually a real phenomenon. From my personal observations and discovery (not scientific), it seems that the reason people are able to judge the "width" of intervals is because of harmony anyway—either the narrow intervals lie within a critical bandwidth on the basilar membrane and "distance" can be inferred from their direct harmonic interaction, or the wider intervals can be matched to the harmonic sound of a specific melody. In either case the idea of "distance" between notes is simply not a thing that can be directly perceived.

I recently got the full program, but it doesn't seem much different from the demo (besides the inclusion of Absolute Pitch Painter). Does Interval Loader teach other components of relative pitch (like minor versus major chords) after I reach a high enough level, or is that somewhere else in the program?

The full program is different from the demo because the higher levels of each game are "unlocked". Once you're using the full version, you'll be able to get past (for example) level 120 on APA, after which you can add more pitches than just C.